\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  35.0 & +  2.00 x_{1} & +  7.00 x_{2} &   & -4.00 x_{4} & +  1.00 x_{5} & -10.00 x_{6} & -5.00 x_{7} & -6.00 x_{8}\\
 x_{10}   &  -15.0 & -1.00 x_{1} &   & -1.00 x_{3} & +  7.00 x_{4} & -6.00 x_{5} & +  9.00 x_{6} & +  1.00 x_{7} & + 10.00 x_{8}\\
 x_{11}   &  -40.0 & + 10.00 x_{1} & -2.00 x_{2} & -8.00 x_{3} & +  6.00 x_{4} & +  4.00 x_{5} & +  3.00 x_{6} & -3.00 x_{7} & +  7.00 x_{8}\\
 x_{12}   &  40.0 & + 10.00 x_{1} & -3.00 x_{2} &   & -7.00 x_{4} & +  6.00 x_{5} & -6.00 x_{6} & -3.00 x_{7} & -3.00 x_{8}\\
 x_{13}   &  -3.0 & +  5.00 x_{1} &   & -10.00 x_{3} & +  1.00 x_{4} & -2.00 x_{5} &   & +  4.00 x_{7} & +  8.00 x_{8}\\
 x_{14}   &  62.0 & -10.00 x_{1} & -6.00 x_{2} & +  8.00 x_{3} & -4.00 x_{4} & +  7.00 x_{5} & +  1.00 x_{6} & +  9.00 x_{7} & -10.00 x_{8}\\
 x_{15}   &  40.0 & +  3.00 x_{1} &   & +  5.00 x_{3} & -7.00 x_{4} & -5.00 x_{5} & -2.00 x_{6} & +  4.00 x_{7} & -5.00 x_{8}\\
\hline
z    &  0.0 & +  1.00 x_{1} & +  3.00 x_{2} & +  3.00 x_{3} & -4.00 x_{4} & -5.00 x_{5} & -1.00 x_{6} & -4.00 x_{7} & -3.00 x_{8}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,15$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -2.00 y_{9} & +  1.00 y_{10} & -10.00 y_{11} & -10.00 y_{12} & -5.00 y_{13} & + 10.00 y_{14} & -3.00 y_{15}\\
 y_{2}   &  1.0 & -7.00 y_{9} &   & +  2.00 y_{11} & +  3.00 y_{12} &   & +  6.00 y_{14} &   \\
 y_{3}   &  1.0  &   & +  1.00 y_{10} & +  8.00 y_{11} &   & + 10.00 y_{13} & -8.00 y_{14} & -5.00 y_{15}\\
 y_{4}   &  1.0 & +  4.00 y_{9} & -7.00 y_{10} & -6.00 y_{11} & +  7.00 y_{12} & -1.00 y_{13} & +  4.00 y_{14} & +  7.00 y_{15}\\
 y_{5}   &  1.0 & -1.00 y_{9} & +  6.00 y_{10} & -4.00 y_{11} & -6.00 y_{12} & +  2.00 y_{13} & -7.00 y_{14} & +  5.00 y_{15}\\
 y_{6}   &  1.0 & + 10.00 y_{9} & -9.00 y_{10} & -3.00 y_{11} & +  6.00 y_{12} &   & -1.00 y_{14} & +  2.00 y_{15}\\
 y_{7}   &  1.0 & +  5.00 y_{9} & -1.00 y_{10} & +  3.00 y_{11} & +  3.00 y_{12} & -4.00 y_{13} & -9.00 y_{14} & -4.00 y_{15}\\
 y_{8}   &  1.0 & +  6.00 y_{9} & -10.00 y_{10} & -7.00 y_{11} & +  3.00 y_{12} & -8.00 y_{13} & + 10.00 y_{14} & +  5.00 y_{15}\\
\hline
z    &  -0 & -35.00 y_{9} & + 15.00 y_{10} & + 40.00 y_{11} & -40.00 y_{12} & +  3.00 y_{13} & -62.00 y_{14} & -40.00 y_{15}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 3 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{11}   &  0.102803738318 & -0.13 y_{9} & -0.01 y_{8} & -0.09 y_{1} & -0.91 y_{12} & -0.54 y_{13} & +  1.03 y_{14} & -0.23 y_{15}\\
 y_{2}   &  1.20560747664 & -7.26 y_{9} & -0.02 y_{8} & -0.19 y_{1} & +  1.19 y_{12} & -1.08 y_{13} & +  8.06 y_{14} & -0.47 y_{15}\\
 y_{3}   &  1.85046728972 & -0.36 y_{9} & -0.17 y_{8} & -0.68 y_{1} & -6.32 y_{12} & +  5.24 y_{13} & +  0.50 y_{14} & -6.21 y_{15}\\
 y_{4}   &  0.18691588785 & -0.06 y_{9} & +  0.71 y_{8} & +  0.10 y_{1} & +  5.90 y_{12} & +  5.20 y_{13} & -4.13 y_{14} & +  3.76 y_{15}\\
 y_{5}   &  0.757009345794 & +  3.67 y_{9} & -0.52 y_{8} & +  0.77 y_{1} & +  3.23 y_{12} & +  1.64 y_{13} & -9.43 y_{14} & +  9.92 y_{15}\\
 y_{6}   &  0.439252336449 & +  4.17 y_{9} & +  0.87 y_{8} & -0.31 y_{1} & +  0.31 y_{12} & +  5.41 y_{13} & -6.61 y_{14} & -3.27 y_{15}\\
 y_{7}   &  1.28037383178 & +  3.92 y_{9} & +  0.07 y_{8} & -0.35 y_{1} & -0.65 y_{12} & -5.21 y_{13} & -6.20 y_{14} & -5.36 y_{15}\\
 y_{10}   &  0.0280373831776 & +  0.69 y_{9} & -0.09 y_{8} & +  0.07 y_{1} & +  0.93 y_{12} & -0.42 y_{13} & +  0.28 y_{14} & +  0.66 y_{15}\\
\hline
z    &  4.53271028037 & -29.86 y_{9} & -1.78 y_{8} & -2.76 y_{1} & -62.24 y_{12} & -24.99 y_{13} & -16.67 y_{14} & -39.39 y_{15}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  29.8598130841 & +  0.13 x_{11} & +  7.26 x_{2} & +  0.36 x_{3} & +  0.06 x_{4} & -3.67 x_{5} & -4.17 x_{6} & -3.92 x_{7} & -0.69 x_{10}\\
 x_{8}   &  1.77570093458 & +  0.01 x_{11} & +  0.02 x_{2} & +  0.17 x_{3} & -0.71 x_{4} & +  0.52 x_{5} & -0.87 x_{6} & -0.07 x_{7} & +  0.09 x_{10}\\
 x_{1}   &  2.75700934579 & +  0.09 x_{11} & +  0.19 x_{2} & +  0.68 x_{3} & -0.10 x_{4} & -0.77 x_{5} & +  0.31 x_{6} & +  0.35 x_{7} & -0.07 x_{10}\\
 x_{12}   &  62.2429906542 & +  0.91 x_{11} & -1.19 x_{2} & +  6.32 x_{3} & -5.90 x_{4} & -3.23 x_{5} & -0.31 x_{6} & +  0.65 x_{7} & -0.93 x_{10}\\
 x_{13}   &  24.9906542056 & +  0.54 x_{11} & +  1.08 x_{2} & -5.24 x_{3} & -5.20 x_{4} & -1.64 x_{5} & -5.41 x_{6} & +  5.21 x_{7} & +  0.42 x_{10}\\
 x_{14}   &  16.6728971963 & -1.03 x_{11} & -8.06 x_{2} & -0.50 x_{3} & +  4.13 x_{4} & +  9.43 x_{5} & +  6.61 x_{6} & +  6.20 x_{7} & -0.28 x_{10}\\
 x_{15}   &  39.3925233645 & +  0.23 x_{11} & +  0.47 x_{2} & +  6.21 x_{3} & -3.76 x_{4} & -9.92 x_{5} & +  3.27 x_{6} & +  5.36 x_{7} & -0.66 x_{10}\\
\hline
z    &  -4.53271028037 & -0.10 x_{11} & -1.21 x_{2} & -1.85 x_{3} & -0.19 x_{4} & -0.76 x_{5} & -0.44 x_{6} & -1.28 x_{7} & -0.03 x_{10}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  29.8598130841 & +  0.13 x_{11} & +  7.26 x_{2} & +  0.36 x_{3} & +  0.06 x_{4} & -3.67 x_{5} & -4.17 x_{6} & -3.92 x_{7} & -0.69 x_{10}\\
 x_{8}   &  1.77570093458 & +  0.01 x_{11} & +  0.02 x_{2} & +  0.17 x_{3} & -0.71 x_{4} & +  0.52 x_{5} & -0.87 x_{6} & -0.07 x_{7} & +  0.09 x_{10}\\
 x_{1}   &  2.75700934579 & +  0.09 x_{11} & +  0.19 x_{2} & +  0.68 x_{3} & -0.10 x_{4} & -0.77 x_{5} & +  0.31 x_{6} & +  0.35 x_{7} & -0.07 x_{10}\\
 x_{12}   &  62.2429906542 & +  0.91 x_{11} & -1.19 x_{2} & +  6.32 x_{3} & -5.90 x_{4} & -3.23 x_{5} & -0.31 x_{6} & +  0.65 x_{7} & -0.93 x_{10}\\
 x_{13}   &  24.9906542056 & +  0.54 x_{11} & +  1.08 x_{2} & -5.24 x_{3} & -5.20 x_{4} & -1.64 x_{5} & -5.41 x_{6} & +  5.21 x_{7} & +  0.42 x_{10}\\
 x_{14}   &  16.6728971963 & -1.03 x_{11} & -8.06 x_{2} & -0.50 x_{3} & +  4.13 x_{4} & +  9.43 x_{5} & +  6.61 x_{6} & +  6.20 x_{7} & -0.28 x_{10}\\
 x_{15}   &  39.3925233645 & +  0.23 x_{11} & +  0.47 x_{2} & +  6.21 x_{3} & -3.76 x_{4} & -9.92 x_{5} & +  3.27 x_{6} & +  5.36 x_{7} & -0.66 x_{10}\\
\hline
z    &  -2.57009345794 & +  0.07 x_{11} & +  3.13 x_{2} & +  3.18 x_{3} & -1.97 x_{4} & -7.34 x_{5} & +  1.92 x_{6} & -3.46 x_{7} & -0.35 x_{10}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  29.8598130841 & +  0.13 x_{11} & +  7.26 x_{2} & +  0.36 x_{3} & +  0.06 x_{4} & -3.67 x_{5} & -4.17 x_{6} & -3.92 x_{7} & -0.69 x_{10}\\
 x_{8}   &  1.77570093458 & +  0.01 x_{11} & +  0.02 x_{2} & +  0.17 x_{3} & -0.71 x_{4} & +  0.52 x_{5} & -0.87 x_{6} & -0.07 x_{7} & +  0.09 x_{10}\\
 x_{1}   &  2.75700934579 & +  0.09 x_{11} & +  0.19 x_{2} & +  0.68 x_{3} & -0.10 x_{4} & -0.77 x_{5} & +  0.31 x_{6} & +  0.35 x_{7} & -0.07 x_{10}\\
 x_{12}   &  62.2429906542 & +  0.91 x_{11} & -1.19 x_{2} & +  6.32 x_{3} & -5.90 x_{4} & -3.23 x_{5} & -0.31 x_{6} & +  0.65 x_{7} & -0.93 x_{10}\\
 x_{13}   &  24.9906542056 & +  0.54 x_{11} & +  1.08 x_{2} & -5.24 x_{3} & -5.20 x_{4} & -1.64 x_{5} & -5.41 x_{6} & +  5.21 x_{7} & +  0.42 x_{10}\\
 x_{14}   &  16.6728971963 & -1.03 x_{11} & -8.06 x_{2} & -0.50 x_{3} & +  4.13 x_{4} & +  9.43 x_{5} & +  6.61 x_{6} & +  6.20 x_{7} & -0.28 x_{10}\\
 x_{15}   &  39.3925233645 & +  0.23 x_{11} & +  0.47 x_{2} & +  6.21 x_{3} & -3.76 x_{4} & -9.92 x_{5} & +  3.27 x_{6} & +  5.36 x_{7} & -0.66 x_{10}\\
\hline
z    &  -2.57009345794 & +  0.07 x_{11} & +  3.13 x_{2} & +  3.18 x_{3} & -1.97 x_{4} & -7.34 x_{5} & +  1.92 x_{6} & -3.46 x_{7} & -0.35 x_{10}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{14} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  44.8886310905 & -0.80 x_{11} & -0.90 x_{14} & -0.10 x_{3} & +  3.78 x_{4} & +  4.83 x_{5} & +  1.79 x_{6} & +  1.67 x_{7} & -0.94 x_{10}\\
 x_{8}   &  1.81438515081 & +  0.01 x_{11} & -0.00 x_{14} & +  0.17 x_{3} & -0.70 x_{4} & +  0.55 x_{5} & -0.85 x_{6} & -0.05 x_{7} & +  0.09 x_{10}\\
 x_{1}   &  3.14385150812 & +  0.07 x_{11} & -0.02 x_{14} & +  0.67 x_{3} & -0.01 x_{4} & -0.55 x_{5} & +  0.46 x_{6} & +  0.49 x_{7} & -0.07 x_{10}\\
 x_{12}   &  59.7865429234 & +  1.06 x_{11} & +  0.15 x_{14} & +  6.39 x_{3} & -6.51 x_{4} & -4.62 x_{5} & -1.28 x_{6} & -0.26 x_{7} & -0.89 x_{10}\\
 x_{13}   &  27.2343387471 & +  0.40 x_{11} & -0.13 x_{14} & -5.31 x_{3} & -4.64 x_{4} & -0.38 x_{5} & -4.52 x_{6} & +  6.04 x_{7} & +  0.38 x_{10}\\
 x_{2}   &  2.06960556845 & -0.13 x_{11} & -0.12 x_{14} & -0.06 x_{3} & +  0.51 x_{4} & +  1.17 x_{5} & +  0.82 x_{6} & +  0.77 x_{7} & -0.03 x_{10}\\
 x_{15}   &  40.3596287703 & +  0.17 x_{11} & -0.06 x_{14} & +  6.18 x_{3} & -3.52 x_{4} & -9.37 x_{5} & +  3.65 x_{6} & +  5.72 x_{7} & -0.68 x_{10}\\
\hline
z    &  3.90951276102 & -0.33 x_{11} & -0.39 x_{14} & +  2.98 x_{3} & -0.37 x_{4} & -3.67 x_{5} & +  4.48 x_{6} & -1.05 x_{7} & -0.45 x_{10}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{13} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  44.377020533 & -0.80 x_{11} & -0.90 x_{14} & +  0.02 x_{13} & +  3.87 x_{4} & +  4.83 x_{5} & +  1.87 x_{6} & +  1.56 x_{7} & -0.95 x_{10}\\
 x_{8}   &  2.67103538663 & +  0.02 x_{11} & -0.01 x_{14} & -0.03 x_{13} & -0.85 x_{4} & +  0.53 x_{5} & -1.00 x_{6} & +  0.14 x_{7} & +  0.10 x_{10}\\
 x_{1}   &  6.58235037134 & +  0.12 x_{11} & -0.04 x_{14} & -0.13 x_{13} & -0.59 x_{4} & -0.60 x_{5} & -0.11 x_{6} & +  1.25 x_{7} & -0.02 x_{10}\\
 x_{12}   &  92.5653123635 & +  1.54 x_{11} & -0.01 x_{14} & -1.20 x_{13} & -12.09 x_{4} & -5.08 x_{5} & -6.72 x_{6} & +  7.01 x_{7} & -0.43 x_{10}\\
 x_{3}   &  5.12800349498 & +  0.08 x_{11} & -0.03 x_{14} & -0.19 x_{13} & -0.87 x_{4} & -0.07 x_{5} & -0.85 x_{6} & +  1.14 x_{7} & +  0.07 x_{10}\\
 x_{2}   &  1.74836173001 & -0.13 x_{11} & -0.12 x_{14} & +  0.01 x_{13} & +  0.57 x_{4} & +  1.17 x_{5} & +  0.87 x_{6} & +  0.70 x_{7} & -0.04 x_{10}\\
 x_{15}   &  72.0318916557 & +  0.64 x_{11} & -0.21 x_{14} & -1.16 x_{13} & -8.91 x_{4} & -9.81 x_{5} & -1.60 x_{6} & + 12.75 x_{7} & -0.23 x_{10}\\
\hline
z    &  19.1983398864 & -0.11 x_{11} & -0.46 x_{14} & -0.56 x_{13} & -2.97 x_{4} & -3.88 x_{5} & +  1.95 x_{6} & +  2.34 x_{7} & -0.24 x_{10}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  49.3986842105 & -0.77 x_{11} & -0.91 x_{14} & -0.04 x_{13} & +  2.28 x_{4} & +  5.84 x_{5} & -1.88 x_{8} & +  1.82 x_{7} & -0.75 x_{10}\\
 x_{6}   &  2.68157894737 & +  0.02 x_{11} & -0.01 x_{14} & -0.03 x_{13} & -0.85 x_{4} & +  0.54 x_{5} & -1.00 x_{8} & +  0.14 x_{7} & +  0.11 x_{10}\\
 x_{1}   &  6.28947368421 & +  0.12 x_{11} & -0.04 x_{14} & -0.12 x_{13} & -0.50 x_{4} & -0.65 x_{5} & +  0.11 x_{8} & +  1.24 x_{7} & -0.04 x_{10}\\
 x_{12}   &  74.5328947368 & +  1.41 x_{11} & +  0.03 x_{14} & -0.99 x_{13} & -6.38 x_{4} & -8.68 x_{5} & +  6.75 x_{8} & +  6.07 x_{7} & -1.14 x_{10}\\
 x_{3}   &  2.84473684211 & +  0.06 x_{11} & -0.02 x_{14} & -0.16 x_{13} & -0.15 x_{4} & -0.53 x_{5} & +  0.85 x_{8} & +  1.02 x_{7} & -0.02 x_{10}\\
 x_{2}   &  4.09078947368 & -0.12 x_{11} & -0.13 x_{14} & -0.02 x_{13} & -0.17 x_{4} & +  1.64 x_{5} & -0.88 x_{8} & +  0.82 x_{7} & +  0.05 x_{10}\\
 x_{15}   &  67.7289473684 & +  0.61 x_{11} & -0.20 x_{14} & -1.11 x_{13} & -7.55 x_{4} & -10.67 x_{5} & +  1.61 x_{8} & + 12.52 x_{7} & -0.40 x_{10}\\
\hline
z    &  24.4144736842 & -0.07 x_{11} & -0.48 x_{14} & -0.62 x_{13} & -4.63 x_{4} & -2.84 x_{5} & -1.95 x_{8} & +  2.61 x_{7} & -0.04 x_{10}\\
\end{array}\]


 $ x_{7} $ enters and Unbounded Dictionary!


 $ x_{7} $ enters and Unbounded Dictionary!
\end{document}
